To what extent can a linear analysis predict the behaviour of a flute model ?

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To what extent can a linear analysis predict the behaviour of a flute model ?

Soizic Terrien, Christophe Vergez, Benoît Fabre

To cite this version:

Soizic Terrien, Christophe Vergez, Benoît Fabre. To what extent can a linear analysis predict the

behaviour of a flute model ?. International Symposium on Musical Acoustics, Jul 2014, Le Mans,

France. pp.171-176. �hal-01038010�

To what extent can a Linear Analysis predict the Behaviour of a Flute Model?

Soizic Terrien ¹⁾ , Christophe Vergez ¹⁾ , Benoˆıt Fabre ²⁾

1) LMA, CNRS, UPR 7051, Aix-Marseille Univ, Centrale Marseille, F-13402 Marseille Cedex20, France.

2) LAM, Sorbonne Universit´es, UPMC Univ Paris 06, CNRS, UMR 7190 Institut Jean Le Rond d’Alembert, 11 rue de Lourmel F-75015 Paris, France.

Abstract

Although they have been widely studied for years, some aspects of the behaviour of flute-like musical instruments remain poorly understood. The study of a physical model of the instrument has demonstrated its interest in the understanding of various phenomena, such as the hysteresis related to regime changes or the variations of the frequency with the blowing pressure. As it involves both nonlinear and delayed terms, an indepth study of the state of the art flute model requires specific numerical methods, which are often computationally expensive. The simplification of the model through its linearisation around a non-oscillating trivial solution is thus particularly interesting, due to the simplicity of the calculations.

The information provided by such an analysis in terms of oscillation frequency or oscillation thresholds of the different periodic solutions has been highlighted in previous work . Surprisingly enough, the present study shows that this simple linear analysis provides information about the stability zones of the different periodic solutions (i.e. the different registers), and allows to predict, in some cases, the register resulting from a transient of the mouth pressure. Such information can be obtained without solving the nonlinear equations and without computing the steady-state oscillations of the model.

1 Introduction

Various studies have highlighted the complex be- haviour of flute-like instruments, and the valuable information arising from the study of a physical model of this kind of instruments [2, 5, 3]. How- ever, even the simplest model involves both nonlin- ear, non smooth and delayed terms [1]. These par- ticular features make both its time-domain integra- tion and its analysis in terms of a (neutral) non- linear dynamical system particularly complex, and costly in terms of computation time [3]. The dras- tic reduction of the model, through its linearisation around a trivial non oscillating (i.e. static) solu- tion thus presents an obvious interest, due to the simplicity of calculations. It is theoretically known that this linear analysis of the model provides in- formation about the stability properties of the con- sidered static solution, and thus, in some cases, about the oscillation threshold (see for example [10]). Moreover, some studies have stressed the ability of this method to provide a rough approx- imation of the oscillation frequency and regime change thresholds [2].

This paper focuses on the information provided by the linear analysis of the state-of-the-art phys- ical model for flute-like instruments in the case of attack transients of the mouth pressure. Such at- tack transients correspond to strongly non station- nary evolutions of the parameters, and to oscilla- tions outside the vinicity of the static solution in- volved in the linearisation. As the linearisation in- volves hypothesis of both quasistatic variations of the parameters, and small perturbations around the static solution, attack transients fall, at first sight, outside the scope of the considered linear analysis. However, due to their musical impor- tance and to the fact that they are directly con- trolled by the instrumentalist, their study is par- ticularly interesting.

The state-of-the-art physical model for flute-like

instruments is first presented in section 2, fol-

lowed by details on its linearisation and analysis

in section 3. Finally, section 4 presents the com-

parison between the results of linear analysis and

time-domain simulations of the complete nonlinear

model. Information provided by the linear analysis

are discussed in terms of oscillation regime reached

during an attack transient, stability ranges of the different periodic regimes, and duration and spec- tral nature of the transients.

2 Model for flute-like instru- ments

2.1 General mechanism of sound pro- duction

Although transverse flutes, recorders, or organ pipes present important differences, the general mechanism of sound production can be, in all cases, described as follows [1]. When the musi- cian blows in the instrument, a naturally unstable jet is created at the channel exit of height h (see figure 1). This jet oscillates around the sharp edge called labium, which constitutes the exciter, pro- viding energy to the resonator, formed by the air column contained in the pipe. The acoustic field thus created in the resonator perturbs in turn the jet at the channel exit. As the jet is naturally unstable, this perturbation is amplified while con- vected along the jet of length W from the channel exit to the labium, thus sustaining the oscillations of the jet around the labium. This auto-oscillation process can be represented through a feedback loop system shown in figure 2.

Figure 1: Recorder section and simplified represen- tation of its exciter, constituted by the oscillation of an unstable air jet around a sharp edge called labium.

2.2 State-of-the-art physical modeling This mechanism of sound production is modeled through a set of equations, each of them being re- lated to an element of the feedback loop system sketched in figure 2. The principal phenomena and the equations associated to each of these elements

Figure 2: Basic modeling of sound production mechanism in flute-like instruments, as a feedback loop system [1].

are briefly recalled in this section; for more details on the complete model, the reader is referred to [1, 2, 3].

Following the empirical model proposed by de la Cuadra [5], the initial transversal perturbation η ₀ (t) of the jet is provided at the channel exit by the acoustic velocity v ac (t). Rayleigh’s theory is applied to describe in a simplified way the convec- tion and amplification of this perturbation along the jet, leading to the transversal displacement at the labium:

η(t) = η ₀ (t − τ )e ^α

^W = h

U _j v _ac (t − τ )e ^α

^W , (1) with U _j the centerline velocity of the unstable jet (directly related to the pressure P _mouth in the mu- sician’s mouth, through the Bernoulli relation), and α _i ≈ ^0.4

h an empirical amplification coeffi- cient [5]. The delay τ , due to the convection du- ration, is related both to the distance W , and to the convection velocity c _v of the perturbation, with 0.3U j < c _v < 0.5U j [4, 5]. It is worth noting that τ is directly related to both the jet velocity U _j hence to the mouth pressure P _mouth (the higher P _mouth , the smaller the delay τ ).

The oscillation of the jet induces an alternative flow injection on each side of the labium (inside and outside the pipe). Following the jet-drive model [6], these two localised flow sources in phase op- position, separated by a distance δ _d , constitute a dipolar pressure source ∆p _src (t).

Moreover, the phenomenon of vortex shedding at the labium [7] gives rise to energy loss modeled through an additional term ∆p los (t) in the source equation.

The pressure source ∆p(t) = ∆p src (t) + ∆p los (t)

exciting the resonator is finally written as:

∆p(t) = ρδ _d bU _j W

d dt

tanh

η(t) − y ₀ b

− ρ 2

v _ac (t) α _vc

2

sgn((v ac (t)),

(2)

with ρ the air density, b the half width of the Bick- ley profile of the jet, y ₀ the offset between the labium and the jet centerline (see figure 1), sgn the sign function, and where α vc represents the vena contracta contraction coefficient, estimated for a sharp edge at 0.6.

The acoustical frequency response of the air col- umn to the pressure source is given by its admit- tance Y (ω) = ^V _∆P

^(ω) _(ω) , which is modelled through a modal decomposition:

V ac (ω) =

"

a ₀ jωb ₀ + c ₀ +

p

X

k=1

a _k jω ω ² _k − ω ² + jω _Q ^ω

k

#

∆P (ω), (3) with ω the pulsation, a _k , ω _k and Q _k the modal amplitude, the resonance pulsation and the qual- ity factor of the k ^th resonance mode (respectively), and a ₀ , b ₀ and c ₀ the coefficients of the so-called uniform mode at zero-frequency. Such coefficients are estimated, for different fingerings, from the geometrical dimensions of a Bressan Zen-On alto recorder, using the software WIAT [8].

The model is finally defined by equations 1, 2, and 3. For sake of numerical conditionning, the re- sulting system is made dimensionless, defining the following variables: ˜ t = ω ₁ t and ˜ v(˜ t) = ^he _bU

j

v _ac (t) (see [3] for more details).

3 Linearisation around the equi- librium

The model described in section 2 can be rewritten as a neutral delayed nonlinear dynamical system [2, 3]:

˙

x (˜ t) = f ( x (˜ t), x (˜ t − τ ˜ ), x ˙ (˜ t − τ ˜ ), λ). (4) where λ is the set of parameters, and x the vector of state variables, constituted by the projections of

˜

v(˜ t) on each mode of the admittance (see equation 3), and their first derivative with respect to time (see [3] for more details).

Due to its neutral delayed nature, the complete resolution and analysis of the model is particularly

complex [9, 3]. Especially, time-domain simula- tions require very high sampling frequency (typ- ically 1 MHz) to provide accurate results, and are thus costly in terms of computation time.

The study of the corresponding linearised system around a non oscillating solution is thus particu- larly interesting, due to the simplicity of the cal- culations.

The model studied here has a trivial non oscillat- ing solution f (0, 0, 0) = 0. Linearisation of system 4 around this static solution leads to the following equation:

˙

x (˜ t) = A ₁ x (˜ t) + A ₂ x (˜ t − τ ˜ ) + A ₃ x ˙ (˜ t − τ ˜ ), (5) where A _i denotes the partial derivative of vector function f with respect to its i ^th argument, evalu- ated at the static solution 0 (see for example [10]).

In this expression, terms A ₂ and A ₃ are directly related to the neutral nature of the system. In a more classical system of ordinary differential equa- tions x ˙ (t) = f ( x (t), λ), modeling for example reed instruments, brass instruments and bowed string instruments, A ₂ = A ₃ = 0.

The stability properties of the static solution are then determined by the roots κ of the characteristic equation associated to equation 5:

det(κI − A ₁ − A ₂ e ^{−κ˜ τ} − A ₃ κe ^{−κ˜ τ} ) = 0, (6) where I represents the identity matrix [10]. In the present case, this equation is solved with the soft- ware DDE-Biftool and its extension for neutral dy- namical systems [11, 9, 12]. The considered static solution remains locally stable as long as all the roots have negative real parts. Conversely, if at least one of the roots has a positive real part, any small disturbance is amplified with time, and the static solution is thus locally unstable. This anal- ysis thus allows to determine the Hopf bifurcation points at which the real parts of two complex con- jugate roots κ become positive. Such a Hopf bifur- cation corresponds to the birth of a periodic solu- tion (and thus to an oscillation threshold), whose frequency at this threshold is driven by the imag- inary part of the considered roots. Each periodic solution is thus associated to a given instability of the static solution.

As an example, figure 3 represents real parts of

the roots of equation 6, with respect to the dimen-

sionless delay ˜ τ (along this paper, the roots are

represented with respect to ˜ τ for sake of readibil- ity and consistency with the numerical resolution method). The modal coefficients correspond to the G ₄ fingering of an alto recorder. It highlights that for the range of ˜ τ represented, the static solution remains unstable. More precisely, two different instabilities (corresponding to two periodic solu- tions) exist for 0.98 < τ < ˜ 1.3. The first instability disapears for ˜ τ slightly below 0.2. According to the imaginary part of the root associated to this insta- bility, it corresponds to a periodic regime at a fre- quency close to the first resonance frequency, the so-called first register. In the same way, the sec- ond instability (corresponding to the second reg- ister) disapears around ˜ τ = 0.1 . At ˜ τ = 0.98,

˜

τ = 0.7 and ˜ τ = 0.55, three additional periodic so- lutions emerge, which respectively correspond to the third, the fourth and the fifth registers, and die out at ˜ τ = 0.05, ˜ τ = 0.04 and ˜ τ = 0.02.

In addition to the prediction of oscillation thresholds and oscillation frequency at each of these thresholds, some previous studies have high- lighted that such a linear analysis can provide a rough approximation of both the regime change thresholds and the oscillation frequency far from oscillation thresholds (through the imaginary parts of the different roots, not represented here) [2].

The present study focuses on the information pro- vided by this linear analysis in the case of attack transients of the mouth pressure.

4 Results: Linear analysis and attack transients

The results of linear analysis are compared to time-domain simulations of the complete nonlin- ear model. Since attack transients correspond both to highly non stationary evolutions of the param- eters, and to oscillations outside the vinicity of the static solution around which the system is lin- earised (equation 5), nothing guarantees that this method can provide valuable information.

4.1 P mouth steps : arrival oscillation regime

For modal coefficients corresponding to the G ₄ fin- gering of an alto recorder, different steps of the mouth pressure between a low value of P _mouth (for which the static solution is stable) and a variable target value P _t of the mouth pressure are carried out through time domain simulations of the model

presented in section 2. A Runge-Kutta solver of or- der 3, implemented in Matlab - Simulink, has been used. Figure 4 represents, with respect to time, the mouth pressure, the acoustic velocity v _ac (t) and the oscillation frequency, for two different values of P _t (P t = 300 Pa and P _t = 400 Pa). The com- parison of the fundamental frequencies highlights that the step with P _t = 300 Pa leads to oscillation on the first register, whereas the step with P _t = 400 Pa leads to oscillation on the second register.

0.2 0.4 0.6 0.8 1 1.2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

dimensionless delay τ ⋅ ω₁

Re(λ)

Pmouth = 300 Pa Pmouth = 400 Pa

instability corresponding to the second register

instability corresponding to the first register

Figure 3: G ₄ fingering: real parts of the roots κ (linearised model), with respect to the dimension- less delay ˜ τ . Each (positive) branch of ℜ (κ) cor- responds to an instability of the static solution, related to a periodic solution. The cross and the circle represent the largest value ℜ (κ), for values of P _t (300 Pa and 400 Pa) leading respectively in time-domain simulation to the first and the second register.

Examining the roots κ at these two points sug- gests that the arrival regime could be driven by the root with the largest real part. In figure 3, which represents roots κ with respect to ˜ τ , the blue cross and the red circle highlight the root with the largest real part, respectively for the two values of the dimensionless delay ˜ τ corresponding to P _t = 300 Pa and P _t = 400 Pa. In that case, the first register is obtained when the root with the largest real part corresponds to the instability giving rise to the first register, whereas the second register is obtained when the root with the largest real part is related to the existence of the second register.

This assumption is confirmed by the realisation

of different steps of the mouth pressure on a wider

range of P _t : as highlighted in figure 5, which repre-

sents the same kind of data as figure 3, the oscilla-

0 0.1 0.2 0.3

−5 0 5

oscillation amplitude

0 0.1 0.2 0.3

0 0.1 0.2 0.3

0 200 400

Mouth pressure (Pa)

0 0.1 0.2 0.3

0 0.1 0.2 0.3

0 500 1000

time (s) oscillation frequency (Hz)

0 0.1 0.2 0.3

time (s)

Figure 4: Time-domain simulation of the complete model, for two steps of P _mouth with a target pres- sure P t = 300 Pa (left) and P t = 400 Pa (right).

Representation of the mouth pressure (high), the acoustic velocity v _ac (t) (middle) and the funda- mental frequency (bottom), with respect to time.

tion regime reached corresponds, in all the tested cases, to the register related to the root with the largest real part.

Through a linear analysis of the model, it is thus possible to predict the range of mouth pressure of the attacks (controlled by the musician) leading to each oscillation regime. Thus, it is possible to bring out some characteristics of a given fingering, such as for example the inhability (or at least the difficulty) to attack on a specific register.

4.2 Stability ranges of the periodic regimes

According to these first results, the analysis of the system linearised around its static solution allows to predict the oscillation regime resulting from a step of the mouth pressure. It thus suggests that the periodic regime related to a given instability of the static solution is necessary stable as long as the associated root is the largest one, in terms of real part. The different periodic solutions branches of the complete model and their stability proper- ties are calculated in the software DDE-Biftool, as described in [3]. Figure 6 represents the roots κ computed for the G ₄ fingering as a function of ˜ τ (already represented in figures 3 and 5), superim- posed with the range of stability of the different corresponding periodic solution branches (i.e. the different registers). It highlights that for each of the five registers, if the associated root κ is the one

0 0.2 0.4 0.6 0.8 1 1.2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

dimensionless delay τ ⋅ ω 1

Re(λ)

arrival regime: 1^st register arrival regime: 2^nd register arrival regime: 3^rd register arrival regime: 4^th register arrival regime: 5^th register

Figure 5: Lines: real parts of roots κ (linearised model), as function of the dimensionless delay ˜ τ , for the G ₄ fingering. Markers: time-domain simu- lation of steps of P _mouth ; representation of the ob- served oscillation regime as function of the value

˜

τ _c corresponding to the step amplitude P _t .

with largest real part, the regime is stable. How- ever the reciprocal is false.

If it does not allow to predict the precise range of stability of a given periodic regime, we conjecture that this analysis nevertheless provides a minimal range of stability for each periodic regime.

4.3 Duration and type of the transients As real parts of roots κ characterise the amplifi- cation, with respect to time, of small disturbances superimposed on the static solution, it is expected that the duration of the transient of v _ac (t) depends on the real part of the root associated to the arrival regime (see for example [10]). For simulations of different steps of the mouth pressure, the duration of the transient of v ac (t) has been compared to the largest real part of the roots κ. The duration of the transient is defined as the time between the instant at which the step of P _mouth occurs and the time at which the oscillation amplitude of v _ac (t) differs by less than 10 percent from the steady-state oscilla- tion amplitude. The results, represented in figure 7 for modal coefficients corresponding to the A ₄ fingering of an alto recorder, show good agreement with the theory, in the sense that the larger the real part of the root, the shorter is the transient.

However, two specific points, for ℜ (κ) = 0.05 and ℜ (κ) = 0.07, show unexpected behaviours.

A more careful study of this two points high-

lights particularly interesting phenomenon: the

0 0.2 0.4 0.6 0.8 1 1.2 0

0.05 0.1 0.15

dimensionless delay τ ⋅ ω₁

Re(λ)

roots of the linearised system 1^st register: stability range 2^nd register: stability range 3^rd register: stability range 4^th register: stability range 5^th register: stability range

Figure 6: Dotted lines: real parts of roots κ (lin- earised model), as function of the dimensionless delay ˜ τ , for the G ₄ fingering. Solid lines: stability range of the periodic solution associated to each instability of the static solution (i.e. branch of ℜ (κ)).

(long) transients of the temporal signals v _ac (t), represented in windows (a) and (b) in figure 7, are in both cases constituted by a quasiperiodic regime. Moreover, examining the real parts of roots κ at these points shows, as highlighted in window (c) in figure 7, that in both cases, two

”branches” of roots (corresponding to two differ- ent periodic solutions) intersect. Case (a) corre- sponds to the intersection between the branch re- lated to the first register and the branch related to the second register. A spectral analysis of the quasiperiodic transient observed at this point high- lights that its two base frequencies correspond to those of the 1 ^st and 2 ^nd registers.

Similarly, case (b) is located at the intersection point between the roots leading respectively to the 2 ^nd and the 4 ^th register (see (c)). As for the case (a), the spectral analysis of the quasiperiodic tran- sient allows to identify the two base frequencies as frequencies of the 2 ^nd and 4 ^th registers.

At these points, it thus seems that the system

”hesitates” between the two instabilities (leading to two different registers). As each of these insta- bilities involves a particular frequency, it results in a long quasiperiodic transient, constituted by these two frequencies.

It results that the computation of the roots κ not only provides an estimation of the relative du- rations of the transients for different values of the mouth pressure (the transient will all the shorter

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

largest real part of the roots κ duration of the transient of vac (s)

1^st register 2^nd register 4^th register 5^th register

(a) (b)

(c)

Figure 7: Time-domain simulation of mouth pres- sure steps, for the A ₄ fingering. Transient duration of v ac with respect to the largest value ℜ (κ) at the value of ˜ τ _c corresponding to the amplitude P _t of the step. (a) and (b): temporal signal v _ac with re- spect to time for the two pathological points. (c):

ℜ (κ) of the linearised model, with respect to ˜ τ . than the real part of the root is large), but also, in some cases, information about the nature of the transient. This result suggests that it would be possible to access, through a comparison of the real parts of the different roots, an estimation of the spectral content of the transient of v _ac . How- ever, this remains to be confirmed.

4.4 Influence of the rise time of P mouth

Previous sections have bring out the interesting connections between the behaviour of the complete (nonlinear) model and the roots of the linearised system, in the case of attack transients formed by steps of the mouth pressure. However, one can question the validity of these results in the case of more realistic transients. In a study focusing on the analysis of attack transients realised by recorder players, Garc´ıa [13] noted rise times of the blowing pressure between 10ms and 40ms.

To test the influence of the attack time, same kind of simulations as in section 4.1 have been achieved with linearly increasing attack transients of P _mouth , with rise time from 10ms to 40ms. If such profile of P _mouth remains a rough represen- tation of real attack transients (mainly because of the linear increase during the attack), they are nev- ertheless less caricatural than the steps previously studied.

As in section 4.1, figure 8 presents the compar-

ison between the oscillation regime resulting from attack transients with a rise time of 40ms (with different values of the target pressure) and real parts of the roots of the linearised system. As previously, it highlights that the arrival oscillation regime is the register resulting from the instability of the static solution associated to the root with the largest real part, and that this arrival regime can thus be predicted by the simple examination of the roots κ. Same results have been observed for different values of the rise time of P _mouth be- tween 0ms and 40ms. If the influence of the attack transient profile remains to be studied, these re- sults already argue in favor of the validity of this conjecture in the case of more realistic transients attack.

0 0.2 0.4 0.6 0.8 1 1.2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

dimensionless delay τ ⋅ ω₁

Re(λ)

arrival regime: 1^st register arrival regime: 2^nd register arrival regime: 3^rd register arrival regime: 5^th register

Figure 8: Lines: real parts of roots κ (lin- earised model), as function of ˜ τ , for the G ₄ fin- gering. Markers: time-domain simulation of steps of P _mouth with rise time of 40ms; representation of the arrival register as function of the value ˜ τ c

corresponding to the step amplitude P _t .

5 Conclusion

This study has presented some comparisons be- tween the analysis of the linearised state-of-the-art model for flutes, and time-domain simulations of attack transients of the complete nonlinear model.

Theoretically, attack transients fall totally out of the scope of linear analysis around the static so- lution. However, very surprisingly, it appears that the roots computed from the linearised model pro- vide valuable information on attack transients of the complete nonlinear model. Indeed, the study of the real parts of the roots is enough to predict the oscillation regime resulting from a step of the mouth pressure, a minimal range of stability of

each periodic regime, and in some cases, a qual- itative estimation of the relative transients dura- tion and spectral content. It is interesting to note that these conjectures seem to remain valid for rise times of the mouth pressure corresponding to those observed on musicians. These results thus allow, in some cases, to predict the model behaviour with- out solving the complete model, ans thus consid- erabily reducing the computation time and com- plexity. To the author knowledge, these surpris- ing results have never been observed before, nei- ther on neutral dynamical systems, nor on simpler ordinary differential systems (modelling for exam- ple reed, brass and bowed-string instruments). In order to test the validity of these conjectures for more realistic attack transients, it would be inter- esting to study, in addition to the rise time of the mouth pressure, the influence of its temporal pro- file. Indeed, only linear profiles of the mouth pres- sure have been considered here. Moreover, since this study suggests the existence of a relation be- tween the quasiperiodic nature of a transient and the roots of the linearised model, it would be in- tereseting to explore in more depth the ability of the method to predict the spectral content of the attack transients.

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